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Integer and fractional packing of graph families

Raphael Yuster

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Definitions and notations

• Let F be any (finite or infinite) family of finite graphs.

The F-packing number of a graph G, denoted ν(F,G), is the maximum number of edge disjoint elements of F in G.

• The fractional version is denoted ν*(F,G).Trivially, ν*(F,G) ≥ ν(F,G).

• Example: ν*({K3} , K4) = 2 ν ({K3} , K4) = 1

• It is NP-Hard to compute ν(F,G) even when F is a single (nontrivial) graph, say K3.

• Computing ν*(F,G) can be done in polynomial time if F is finite (LP).

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Main result and its algorithmic consequences

• Our main result: If G has n vertices then

ν*(F,G) - ν(F,G) =o(n2).

• Corollary:ν(F,G) can be approximated to within an o(n2) additive factor in polynomial time.

• Our proof supplies a randomized algorithm that finds a packing of size ν(F,G) – o(n2) in polynomial time. The degree of the polynomial depends only on F if F is finite.

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Previous research

• The case where F={H} was first proved by Haxell and Rödl (Combinatorica 2001).

Their 25 page proof is very difficult. The main problem lies in the fact that their method requires proving that there is a fractional packing which is only slightly less than optimal, and which assigns to every copy of H either 0 or a value greater than τ for some τ > 0 which is only a function of H.

There does not seem to be an easy way to generalize their proof to the “family” case.

• Our proof is mostly probabilistic and is much simpler than the HR proof, and can easily be adapted to the “family” case.

• For “simplicity” we shall only demonstrate our proof for F={Kk}.

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Tools used

• The regularity lemma:

For every γ>0, there is an integer M(γ)>0 such that for every graph G of order n > M there is a γ-regular partition of the vertex set of G into m classes, for some 1/γ < m < M.

• A “randomlike behavior lemma”

Let δ and ζ be positive reals.There exist γ = γ(δ, ζ, k) and T=T(δ, ζ, k)such that the following holds.Let W be a k-partite graph with vertex classes V1,…,Vk and |Vi|=t >T. Furthermore, each (Vi,Vj) is a γ-regular pair with density > δ. Then, there exists a spanning subgraph W' with at least (1-ζ)|E(W)| edges such that for all e E(W'), if e E(Vi,Vj) then

|c(e) - tk-2 (∏d(s,p))/d(i,j)| < ζ tk-2

Where c(e) is the number of Kk containing e

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V1 V2

V3

e

Examplek=3 d(1,3)=d(2,3)= ½

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Tools used – cont.

• The Frankl- Rödl hypergraph matching theorem:

For an integer r > 2 and a real β > 0 there exists a real μ > 0 so that if an r-uniform hypergraph on q vertices has the following properties for some d:

(i) (1- μ)d < deg(x) < (1+ μ)d for all x

(ii) deg(x,y) < μd for all distinct x and y

then there is a matching of size at least(q/r)(1-β).

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The proof

• Let ε > 0. We shall prove there exists N=N(k,ε) such that for all n > N, if G is ann-vertex graph then ν*(Kk,G) - ν(Kk,G) < εn2.

• A consistent “horrible” parameter selection:δ=ε/4β=ε/4μ=μ(β,k2) of Frankl- Rödlζ= 0.5μδk2

γ=γ(δ, ζ ,k) T=T(δ, ζ ,k) as in previous lemmaM=M(γε/k2) as in regularity lemmaN= sufficiently large w.r.t. these parameters

• Fix an n-vertex graph G with n > N vertices. Let ψ be a fractional packing withw(ψ)= ν*(Kk,G) = αn2> εn2.

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The proof cont.

• Apply regularity lemma to G and obtain aγ‘-regular partition with m' parts, whereγ‘=γε/(8k2) and 1/γ' < m' < M(γ').

• Each part has n/m’ vertices but the problem is that there could be many bad copies of Kk with two vertices in the same vertex class (we want to avoid this).

We refine the partition by randomly partitioning each part into 8k2/ε parts.The refined partition is now γ-regular.What we gain: with positive probability the contribution of the still remaining bad copies to w(ψ) is less than εn2/16.

We may now assume that there are no bad copies.

Let V1,…,Vm be the vertex classes of the refined partition.

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The proof cont.

• Let G* be the spanning subgraph of G consisting of the edges with endpoints in distinct vertex classes of the refined partition that form a γ-regular pair with density > δ.

• Let ψ* be the restriction of ψ to G* (namely, to those Kk that “survived”). It is easy to see

ν*)Kk,G*) ≥ w(ψ*) > w(ψ)- δn2 ≥ (α-δ)n2

• Let R denote the m-vertex supergraph.We define a fractional packing ψ' of R as follows:For each H=Kk in R with H={u1,…,uk}

ψ’(H)= w(ψ* | [Vu1,…Vuk]) m2 / n2

important observations:

ψ’ is proper

ν*)Kk,R) ≥ w(ψ’) = w(ψ*)m2/n2 ≥ (α-δ)m2

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1/2

1/3

ExampleThree vertex classes with n/m=5 in G*,with two K3 copies inside them having weights 1/2 and 1/3

The vertices corresponding to these classesin R. The weight of the K3 is (1/2+1/3)/52

1/30

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The proof cont.

• We use ψ' to define a random coloring of the edges of G*. Our “colors” are the copies of Kk in R:

Suppose H is a Kk copy in R that contains the edge (i,j). Each e E(Vi,Vj) is colored “H” with probability ψ’(H)/d(i,j).

– The choices made by distinct edges of G* are independent.

– The random coloring is probabilistically sound as the sum of ψ’(H) taken over all Kk copies H containing (i,j) is at most d(i,j) ≤ 1.

– Some edges might stay uncolored.

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ji

2/25

3/25

Vi Vj

ExampleTwo K3 containing (i,j) with d(i,j)=1/5

In E(Vi,Vj)Prob(- - -) = 2/5

Prob(___ ) = 3/5

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The proof cont.

• Let H be a Kk-copy in R with ψ'(H) > m1-k.W.l.o.g. H={1,…,k}. Let WH=G*[V1,…,Vk].

– WH satisfies the conditions of the randomlike behavior lemma.

– Let W'H be the spanning subgraph of WH whose existence is guaranteed by the lemma.

– Let XH denote the random spanning subgraph of W'H consisting only of the edges whose “color” is “H”.

– For an edge e E(XH), let cH(e) denote the set of Kk copies in XH that contain e.Put r=k(k-1)/2. A crucial lemma is:

• With probability > 1-m3/n, for all e E(XH)

| cH(e) - tk-2 ψ'(H)r-1 | < μ tk-2 ψ'(H)r-1

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The proof cont.

• We also need a lower bound for the number of edges of XH: With probability at least 1-1/n,

|E(XH)| > r(1-2ζ) ψ’(H) n2 / m2

• Since there are at most mk copies of Kk in R we have that with probability at least1-mk/n - mk+3/n > 0all copies H of Kk in R with ψ'(H) > m1-k

satisfy the statements of the last two lemmas. We therefore fix such a coloring.

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The proof cont.

• Let H be a Kk in R with ψ'(H) > m1-k.We construct an r-uniform hypergraph LH:

– The vertices of LH are the edges of XH.

– The edges of LH are the edge sets of the copies of Kk in XH

• Our hypergraph satisfies the FR theorem with d=tk-2 ψ'(H)r-1. (Notice that the co-degree of any two vertices of is less than tk-3)

• By FR we have (q/r)(1-β) edge disjoint Kk in XH. As q > r(1-2ζ) ψ’(H) n2 / m2 we have (1-β) (1-2ζ) ψ’(H) n2 /m2 ≥ (1-2β)ψ’(H)n2 /m2

• Recall that w(ψ') ≥ m2(α-δ). Since thecontribution of copies with ψ’(H) ≤ m1-k tow(ψ') is < m, summing the last inequality over all H with ψ’(H) > m1-k we have at least(α-ε)n2 edge disjoint Kk in G.